All Questions
Tagged with matrix-decomposition positive-definite
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For any SVD $A = U\Sigma V^T$ of a positive definite, symmetric matrix $A \in \mathbb{R}^{n \times n}$, we have $U = V$.
First of all, I've read through all of the answers here and here, but neither of those threads was able to give completely satisfying answers.
Now, I understand that, if $A$ is symmetric and positive ...
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Factorizing $AMA^T+N=WW^T$ efficiently.
This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed.
...
- 6,076
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For an $n$ by $n$ symmetric matrix $M$, why does $Tr(M)^2/Tr(M^2)>n-1$ imply that M is positive or negative definite?
Let $M$ be an symmetric square matrix of size $n$. I am trying to prove that
$Tr(M)^2/Tr(M^2)>n-1$ is a sufficient condition for proving that $M$ is positive or negative definite. If in addition $...
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SVD of "normalized" SPD matrix: $P=\sigma \rho \sigma^T$. Relate SVD of $rho$ to the SVD of $P$?
$P\in\mathbb{R}^{n \times n}$ is a symmetric positive definite (SPD) covariance matrix, and can be factorized into standard deviations and correlations as $P=\sigma \rho \sigma^T=\sigma \rho \sigma$. ...
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Distance between two positive semidefinite matrices
Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure ...
- 562
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Is there a name for this kind of matrix decomposition?
A square matrix $\mathbf{A}$ (of size $a\times a$) is not full rank and I want to decompose it as $\mathbf{A}=\mathbf{B}\mathbf{B}^\top$ where $\mathbf{B}$ is an $a\times(a-1)$ matrix of the $a-1$ ...
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Show that a matrix has a Cholesky factorization providing that it can be written as a product of a matrix and its transpose [duplicate]
$A$ is an invertible real square matrix ($A \in \mathbb{M_{n}(\mathbb{R})}$ and $det(A) \neq 0$).
Let's consider another matrix $B \in \mathbb{M_{n}(\mathbb{R})}$ such that:
$$B = {}^\intercal A \cdot ...
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1
answer
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Cholesky inverse
I have the Cholesky decomposition $LL^T$ of a symmetric positive definite matrix. I then compute a result in the form of $A=LXL^T$, where $A$ and $X$ are also symmetric positive definite matrices.
I ...
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Does $v^Tw>0$ imply that $\exists A>0$ such that $v=Aw$?
Let $v,w \in \mathbb{R}^n$ satisfy $v^Tw>0$.
Does there exist a symmetric positive-definite matrix $g$ such that $v=gw$?
The condition $v^Tw>0$ is necessary for the existence of such $g$:
$$
v^...
- 25.3k
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Finding the positive projection in the frobenius norm
I am trying to find the positive projection in the frobenius norm of a real matrix. Consider the following matrix $\hat{Z}$:
\begin{bmatrix}1&2&0\\0&5&0\\0&8&9\end{bmatrix}
I ...
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Inequality for PSD block matrices
Given a positive definite matrix $M\in\mathbb{R}^{(2n)\times(2n)}$ of the following block form
$$M = \begin{bmatrix}A & X\\X^\top& B\end{bmatrix},$$
where $A, B\in\mathbb{R}^n$ are positive ...
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Decomposition of positive definite matrix.
Let $A$ be a positive definite matrix. Following are the decomposition's of matrix $A$:
$A = PP$
$A = MM^{T}$
Empirically we found that $||P||_{F}^2 = ||M||_{F}^2$ but how to prove this ...
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Conditions for positivity of this symmetric real matrix
I have the following real symmetric matrix $M$ of size 3:
\begin{align}
M =
\begin{pmatrix}
a & b & c \\
b & d & e \\
c & e & f
\end{pmatrix},
\end{align}
for real parameters ...
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An Easy Looking Positive Semidefinite Matrices Implication
I'm reading the article "Controlling the false discovery rate via knockoffs" by Candes and Barber (https://arxiv.org/abs/1404.5609) and faced the following problem that I couldn't handle.
We ...
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Efficient algorithm of rank-one update of the Cholesky decomposition
Suppose that I have a symmetric positive definite matrix $X$ and that I Cholesky-decompose it
$$ X = L L^T $$
Now, given a vector $v$, suppose we want to decompose the following matrix using the ...
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